The dangerous part is the doorways, not the drive
A small bird loses heat astonishingly fast, and the riskiest moments of a winter trip are not the miles on the road — they are the thirty seconds between your front door and the car, and again at the other end, when cold air hits a bird who has just left a warm room. Plan the journey backwards from that fact: the car is warmed up and waiting before the bird leaves the house, the carrier is ready by the door, and someone has already cleared the path so there is no fumbling with keys while a chilled bird waits in a cold carrier on the doorstep. If the appointment can move, choose the mildest part of the day; if it cannot, shorten every outdoor second.
A small carrier beats his cage, every time
Travel him in a small, solid carrier rather than his cage. A cage is mostly open bars: it lets moving air straight through, rattles, and gives a panicking bird room to fly into the sides. A carrier holds a pocket of still air, sits low and stable, and feels like a hiding place instead of an exposure. Line the floor with a towel he can grip rather than slide on, add a low perch only if he is a confident traveller — many small birds do better on the padded floor — and drape a blanket or thick towel over the whole carrier, leaving a breathing gap at one end. The cover does three jobs at once: it blocks drafts, holds warmth in, and keeps the passing world out of his sight. For a longer drive, a warm — not hot — water bottle wrapped in a towel and placed against the outside of one carrier wall gives gentle radiant warmth without anything hot touching the bird.
In the car: warm, still, and boring
Run the car until the interior is genuinely warm before he comes out, and keep the heating steady for the whole trip. Strap the carrier on a back seat with the seatbelt so it cannot tip or slide, positioned away from any vent blowing directly at it — warm air aimed at the cover is fine, a jet of it through the carrier is the draft you are trying to avoid. Drive smoothly, keep music low, and talk to him occasionally in your normal voice. At any stop, remember the cabin cools fast once the engine is off: keep stops to a few minutes, park out of the wind, and never leave him sitting in a cooling car while you run other errands on the way. If the drive is long, a brief check by lifting one corner of the cover — looking for a relaxed bird, not a fluffed, shivering one — is all the monitoring he needs.
Arriving without a second shock
At the vet or the new home, carry him straight indoors before any paperwork, and let him sit in the covered carrier in the warm room for a few minutes so the temperature change registers as one step, not two. Uncover him gradually and let him look around from the carrier before any door opens. Trips like this go smoothly when the paperwork is as prepared as the bird: travelling with his KinsLe Pet Passport means his records are together in one portable packet, ready for the vet visit or to hand over with him at his new home.